For the situation you describe, i.e. focal length limited and cropping deeper than DX, the D600 is not an upgrade to D7000.
IMO D300s is still king here due to its frame rate and very deep buffer, and D800 is next best and in some ways better, with a bit better pixel density and AF to f/8. D7000 is close too but a clear 3rd place due to its AF system and shallow buffer. The D600 might well be 4th place on this list, but with D7000 sub-$1K I think 4th place doesn't matter.
So is there an upgrade path from D7000 for DX bird shooters? I think my above list of 1/2/3 are all close enough that the answer is no. If anything it's D800, but only if you have other purposes for which you'll take advantage of the FX sensor. Otherwise join the long waiting list for the theoretical 24MP DX/8fps D400 which we'd all love to have.